For the majority of this paper, we use the Planck 2018 HFI half-mission frequency
maps in Healpix format [1]} at a resolution NSIDE=2048 available from the Planck Legacy Archivehttps://pla.esac.int. (hereafter PLA). The only preprocessing applied to these maps before
applying the transforms REF -REF  is to remove the means within
the unmasked area of sky. For example, for map \(T_i\)  we subtract the mean
\((T^i)_{\rm mean} = \sum _p w^{T_i}_p (T_i)_p / \sum _p w^{T_i}_p, \) 
